Upload app.py

app.py

@@ -0,0 +1,1140 @@
+import streamlit as st
+import numpy as np
+import matplotlib.pyplot as plt
+import tensorflow as tf
+import tensorflow_probability as tfp
+from math import sqrt
+from scipy import stats
+import pandas as pd
+
+tfd = tfp.distributions
+tfl = tfp.layers
+
+st.title("Probability Distributions")
+add_selectbox = st.sidebar.selectbox(
+    'Choose an Option',
+    ('Discrete Univariate', 'Continuous Univariate')
+)
+
+
+
+# st.title("1 dimensional normal distribution")
+
+def Sum(p1val,p2val,zval):
+    l = int(max(max(zval), len(zval)))
+    l=l*2
+    s=[0]*l
+    for i in range(len(zval)):
+        for j in range(len(zval)):
+            s[int(zval[i]+zval[j])]+=p1val[i]*p2val[j]
+    return s
+def Product(p1val,p2val,zval):
+    l=int(max(max(zval),len(zval)))
+    l=l**2
+    s=[0]*l
+    for i in range(len(zval)):
+        for j in range(len(zval)):
+            s[int(zval[i]*zval[j])]+=p1val[i]*p2val[j]
+    return s
+
+def Normal():
+    st.header("Normal distribution")
+    p = tfd.Normal(2, 1)
+    mean = st.slider('Mean', -5, 5, 0)
+    std = st.slider('Scale', 0, 5, 1)
+    z = f"""\\begin{{array}}{{cc}}
+      \mu &= {mean} \\\\
+      \sigma &= {std}
+    \\end{{array}}
+    """
+    st.latex(z)
+    st1 = r'''
+        mean= \[\mu\]
+
+Variance = \[ \sigma ^2\]
+
+Entropy = \[ \frac{1}{2}\log (2 \pi \sigma ^2) + \frac{1}{2} \]
+        '''
+
+    st.latex(st1)
+    q=tfd.Normal(mean,std)
+    z_values = tf.linspace(-5, 5, 200)
+    z_values = tf.cast(z_values, tf.float32)
+    prob_values_p = p.prob(z_values)
+    prob_values_q = q.prob(z_values)
+
+    fig, ax = plt.subplots()
+    ax.plot(z_values, prob_values_p, label=r'Distribution with unknown parameter', linestyle='--', lw=5, alpha=0.5)
+    ax.plot(z_values, prob_values_q, label=r'Distribution with given parameters')
+
+    ax.set_xlabel("x")
+    ax.set_ylabel("PDF(x)")
+    ax.legend()
+    ax.set_ylim((0, 1))
+
+    st.pyplot(fig)
+    kl = tfd.kl_divergence(q, p)
+    st.latex(f"D_{{KL}}(q||p) \\text{{  is : }}{kl:0.2f}")
+
+    st.subheader("Sum of two Normal Distributions")
+    z_values1 = tf.linspace(-10, 10, 21)
+    z_values1 = tf.cast(z_values1, tf.float32)
+
+    mean1 = st.slider('Mean1', -5, 5, 0)
+    std1 = st.slider('Std1', 0, 5, 1)
+    mean2 = st.slider('Mean2', -5, 5, 0)
+    std2 = st.slider('Std2', 0, 5, 1)
+    q1=tfd.Normal(mean1,std1)
+    q2=tfd.Normal(mean2,std2)
+    prob_values_q1 = list(q1.prob(z_values1))
+    prob_values_q2 = list(q2.prob(z_values1))
+
+    fig2, (ax2,ax3) = plt.subplots(1,2)
+    ax2.plot(z_values1, prob_values_q1, label=r'Normal(mean1,std1)')
+    ax3.plot(z_values1, prob_values_q2, label=r'Normal(mean2,std2)')
+
+    ax2.set_xlabel("x")
+    ax2.set_ylabel("PDF(x)")
+    ax2.set_title("Normal(mean1,std1)")
+    ax2.set_ylim((0, 1))
+
+    ax3.set_xlabel("x")
+    ax3.set_ylabel("PDF(x)")
+    ax3.set_title("Normal(mean2,std2)")
+    ax3.set_ylim((0, 1))
+
+    st.pyplot(fig2)
+
+    prob_values_sum = Sum(prob_values_q1, prob_values_q2, z_values1)
+    q3 = tfd.Normal(mean1+mean2, sqrt(((std1)**2 + (std2)**2)))
+    prob_values_q3 = q3.prob(range(len(prob_values_sum)))
+
+    fig3, ax4 = plt.subplots()
+    ax4.plot(range(len(prob_values_sum)), prob_values_sum, label=r'Normal(mean1,std1)+Normal(mean2,std2)', linestyle='--', lw=5, alpha=0.5)
+    ax4.plot(range(len(prob_values_sum)), prob_values_q3, label=r'Normal(mean1+mean2, sqrt((std1^2 + std2^2)')
+
+    ax4.set_xlabel("x")
+    ax4.set_ylabel("PDF(x)")
+    ax4.legend()
+    ax4.set_ylim((0, 1))
+
+    st.pyplot(fig3)
+    st.markdown("Sum of two Normal distributions yields a Normal distribution")
+
+    st.subheader("Relationship between Poisson and Normal Distribution")
+    rate3 = st.slider('lambda1', 100, 500, 250, 50)
+    q4 = tfd.Poisson(rate=rate3)
+    z_values1 = tf.linspace(0, 600, 601)
+    z_values1 = tf.cast(z_values1, tf.float32)
+    prob_values_q4 = list(q4.prob(z_values1))
+    q5 = tfd.Normal(rate3, sqrt(rate3))
+    prob_values_q5 = list(q5.prob(z_values1))
+
+    fig4, ax5 = plt.subplots()
+    ax5.stem(z_values1, prob_values_q4, label=r'Poisson(lambda1)]', linefmt='r', markerfmt='ro')
+    ax5.plot(z_values1, prob_values_q5, label=r'Normal(lamda1,sqrt(lambda1)', lw=3)
+    ax5.set_xlabel("x")
+    ax5.set_ylabel("PDF(x)")
+    ax5.legend()
+    ax5.set_ylim((0, 0.1))
+    st.pyplot(fig4)
+    st.markdown("For large values of lambda, Poisson(lambda) becomes approximately a normal distribution having mean= lambda and variance= lambda")
+
+def Exponential():
+    st.subheader("Exponential distribution")
+    p = tfd.Exponential(2)
+    rate = st.slider('Lambda', 1, 5, 1)
+    cdf = r'''
+            cdf  = $\int_a^b f(x)dx$ 
+            '''
+
+    st.latex(cdf)
+    q = tfd.Exponential(rate)
+    z_values = tf.linspace(-5, 5, 200)
+    z_values = tf.cast(z_values, tf.float32)
+    prob_values_p = p.prob(z_values)
+    prob_values_q = q.prob(z_values)
+
+    fig, ax = plt.subplots()
+    ax.plot(z_values, prob_values_p, label=r'p', linestyle='--', lw=5, alpha=0.5)
+    ax.plot(z_values, prob_values_q, label=r'q')
+
+    ax.set_xlabel("x")
+    ax.set_ylabel("PDF(x)")
+    ax.legend()
+    ax.set_ylim((0, 1))
+
+    st.pyplot(fig)
+    kl = tfd.kl_divergence(q, p)
+    st.latex(f"D_{{KL}}(q||p) \\text{{  is : }}{kl:0.2f}")
+
+    st.subheader("Relationship between Gamma and Exponential Distribution")
+    alpha1 = 1
+    st.write("alpha1=1")
+    beta1 = st.slider('beta1', 0.0, 10.0, 5.0, 0.5)
+    q2 = tfd.Gamma(concentration=alpha1, rate=beta1)
+    prob_values_q2 = list(q2.prob(z_values))
+    q3 = tfd.Exponential(beta1)
+    prob_values_q3 = list(q3.prob(z_values))
+    fig3, ax3 = plt.subplots()
+    ax3.plot(z_values, prob_values_q2, linestyle='--', lw=3, alpha=0.5, label=r'Gamma(1,beta)')
+    ax3.plot(z_values, prob_values_q3, label=r'Exponential(beta)')
+
+    ax3.set_xlabel("x")
+    ax3.set_ylabel("PDF(x)")
+    ax3.legend()
+    # ax3.set_ylim((0, 1))
+    st.pyplot(fig3)
+
+
+def Uniform():
+    st.subheader("Uniform distribution")
+    p = tfd.Uniform(0,1)
+    low = st.slider('low', 0, 5, 1)
+    high = st.slider('high', 1, 6, 1)
+
+    cdf = r'''
+            cdf  = $\int_a^b f(x)dx$ 
+            '''
+
+    st.latex(cdf)
+    q = tfd.Uniform(low,high)
+    z_values = tf.linspace(-5, 5, 200)
+    z_values = tf.cast(z_values, tf.float32)
+    prob_values_p = p.prob(z_values)
+    prob_values_q = q.prob(z_values)
+
+    fig, ax = plt.subplots()
+    ax.plot(z_values, prob_values_p, label=r'p', linestyle='--', lw=5, alpha=0.5)
+    ax.plot(z_values, prob_values_q, label=r'q')
+
+    ax.set_xlabel("x")
+    ax.set_ylabel("PDF(x)")
+    ax.legend()
+    ax.set_ylim((0, 1))
+
+    st.pyplot(fig)
+    kl = tfd.kl_divergence(q, p)
+    st.latex(f"D_{{KL}}(q||p) \\text{{  is : }}{kl:0.2f}")
+    st.subheader("Relationship between Beta Distribution and Uniform Distribution")
+
+    st.write("beta = alpha= 1")
+    q4 = tfd.Beta(1, 1)
+    q5 = tfd.Uniform(0, 1)
+    z_values1 = tf.linspace(0, 1, 200)
+    z_values1 = tf.cast(z_values1, tf.float32)
+    prob_values_q4 = list(q4.prob(z_values1))
+    prob_values_q5 = list(q5.prob(z_values1))
+
+    fig3, ax3 = plt.subplots()
+    ax3.plot(z_values1, prob_values_q4, label=r'Beta(1,1)', linestyle='--', lw=3)
+    ax3.plot(z_values1, prob_values_q5, label=r'Uniform(0,1)')
+
+    ax3.set_xlabel("x")
+    ax3.set_ylabel("PDF(x)")
+    ax3.legend()
+    # ax3.set_ylim((0, 1))
+
+    st.pyplot(fig3)
+
+
+def Cauchy():
+    st.subheader("Cauchy distribution")
+    p = tfd.Cauchy(0, 0.5)
+    loc = st.slider('location', 0.0, 5.0, 1.0, 0.5)
+    sc = st.slider('scale', 0.0, 5.0, 1.0, 0.5)
+
+    cdf = r'''
+                        cdf  = $\int_a^b f(x)dx$ 
+                        '''
+
+    st.latex(cdf)
+    q = tfd.Cauchy(loc, sc)
+    z_values = tf.linspace(-5, 5, 200)
+    z_values = tf.cast(z_values, tf.float32)
+    prob_values_p = p.prob(z_values)
+    prob_values_q = q.prob(z_values)
+
+    fig, ax = plt.subplots()
+    ax.plot(z_values, prob_values_p, label=r'p', linestyle='--', lw=5, alpha=0.5)
+    ax.plot(z_values, prob_values_q, label=r'q')
+
+    ax.set_xlabel("x")
+    ax.set_ylabel("PDF(x)")
+    ax.legend()
+    ax.set_ylim((0, 1))
+
+    st.pyplot(fig)
+    kl = tfd.kl_divergence(q, p)
+    st.latex(f"D_{{KL}}(q||p) \\text{{  is : }}{kl:0.2f}")
+
+    st.subheader("Relationship between Cauchy and StudentT Distribution")
+    q2 = tfd.Cauchy(0,1)
+    q3 = tfd.StudentT(1,0,1)
+    prob_values_q2 = list(q2.prob(z_values))
+    prob_values_q3 = list(q3.prob(z_values))
+
+    fig2, ax2 = plt.subplots()
+    ax2.plot(z_values, prob_values_q2, label=r'Cauchy(0,1)', linestyle='--', lw=3)
+    ax2.plot(z_values, prob_values_q3, label=r'StudentT(1,0,1)')
+
+    ax2.set_xlabel("x")
+    ax2.set_ylabel("PDF(x)")
+    ax2.legend()
+    ax2.set_ylim((0, 1))
+
+    st.pyplot(fig2)
+
+
+def Chi():
+    st.subheader("Chi distribution")
+    p = tfd.Chi(3)
+    d = st.slider('dof', 0.0, 5.0, 1.0, 0.5)
+
+    cdf = r'''
+                        cdf  = $\int_a^b f(x)dx$ 
+                        '''
+
+    st.latex(cdf)
+    q = tfd.Chi(d)
+    z_values = tf.linspace(-5, 5, 200)
+    z_values = tf.cast(z_values, tf.float32)
+    prob_values_p = p.prob(z_values)
+    prob_values_q = q.prob(z_values)
+
+    fig, ax = plt.subplots()
+    ax.plot(z_values, prob_values_p, label=r'p', linestyle='--', lw=5, alpha=0.5)
+    ax.plot(z_values, prob_values_q, label=r'q')
+
+    ax.set_xlabel("x")
+    ax.set_ylabel("PDF(x)")
+    ax.legend()
+    ax.set_ylim((0, 1))
+
+    st.pyplot(fig)
+    kl = tfd.kl_divergence(q, p)
+    st.latex(f"D_{{KL}}(q||p) \\text{{  is : }}{kl:0.2f}")
+
+    #st.subheader("Transformation of Chi Distribution")
+    #d2 = st.slider('dof2', 0, 5, 1, 1)
+    #p1 = tfd.Chi(d2)
+    #prob_values_new = list(p1.prob(z_values))
+    #z_values2 = np.zeros(len(z_values))
+    #for i in range(len(z_values2)):
+     #   z_values2[i] = z_values[i]**2
+
+    #q1 = tfd.Chi2(d2)
+    #new = list(q1.prob(z_values))
+   # fig2, ax2 = plt.subplots()
+  #  ax2.plot(z_values, prob_values_new, label=r'X->Chi(d)', lw=3)
+ #   ax2.plot(z_values2, prob_values_new, label=r'X transformed to X^2', lw=2)
+#    ax2.plot(z_values2, new, label=r'Chi2(d)', linestyle='--', lw=2)
+
+    #ax2.set_xlabel("x")
+    #ax2.set_ylabel("PDF(x)")
+    #ax2.legend()
+    #st.pyplot(fig2)
+
+def Chi_squared():
+    st.subheader("Chi-squared distribution")
+    p = tfd.Chi2(4)
+    dof = st.slider('dof', 0.0, 10.0, 2.0,0.5)
+
+    cdf = r'''
+                        cdf  = $\int_a^b f(x)dx$ 
+                        '''
+
+    st.latex(cdf)
+    q = tfd.Chi2(dof)
+    z_values = tf.linspace(0, 10, 200)
+    z_values = tf.cast(z_values, tf.float32)
+    prob_values_p = p.prob(z_values)
+    prob_values_q = q.prob(z_values)
+
+    fig, ax = plt.subplots()
+    ax.plot(z_values, prob_values_p, label=r'p', linestyle='--', lw=5, alpha=0.5)
+    ax.plot(z_values, prob_values_q, label=r'q')
+
+    ax.set_xlabel("x")
+    ax.set_ylabel("PDF(x)")
+    ax.legend()
+    ax.set_ylim((0, 1))
+
+    st.pyplot(fig)
+    kl = tfd.kl_divergence(q, p)
+    st.latex(f"D_{{KL}}(q||p) \\text{{  is : }}{kl:0.2f}")
+
+    st.subheader("Relationship between Chi-Squared and Exponential Distribution")
+    q2 =tfd.Chi2(2)
+    q3 = tfd.Exponential(0.5)
+    prob_values_q2 = list(q2.prob(z_values))
+    prob_values_q3 = list(q3.prob(z_values))
+
+    fig2, ax2 = plt.subplots()
+    ax2.plot(z_values, prob_values_q2, label=r'Chi-Squared(2)', linestyle='--', lw=3)
+    ax2.plot(z_values, prob_values_q3, label=r'Exponential(0.5)')
+
+    ax2.set_xlabel("x")
+    ax2.set_ylabel("PDF(x)")
+    ax2.legend()
+    ax2.set_ylim((0, 1))
+
+    st.pyplot(fig2)
+
+
+def Laplace():
+    st.subheader("Laplace distribution")
+    p = tfd.Laplace(0, 3)
+    m = st.slider('mu', 0, 5, 1)
+    s = st.slider('sigma', 0, 5, 1)
+
+    cdf = r'''
+                        cdf  = $\int_a^b f(x)dx$ 
+                        '''
+
+    st.latex(cdf)
+    q = tfd.Laplace(m, s)
+    z_values = tf.linspace(-5, 5, 200)
+    z_values = tf.cast(z_values, tf.float32)
+    prob_values_p = p.prob(z_values)
+    prob_values_q = q.prob(z_values)
+
+    fig, ax = plt.subplots()
+    ax.plot(z_values, prob_values_p, label=r'p', linestyle='--', lw=5, alpha=0.5)
+    ax.plot(z_values, prob_values_q, label=r'q')
+
+    ax.set_xlabel("x")
+    ax.set_ylabel("PDF(x)")
+    ax.legend()
+    ax.set_ylim((0, 1))
+
+    st.pyplot(fig)
+    kl = tfd.kl_divergence(q, p)
+    st.latex(f"D_{{KL}}(q||p) \\text{{  is : }}{kl:0.2f}")
+
+
+def Pareto():
+    st.subheader("Pareto distribution")
+    p = tfd.Pareto(2, 1)
+    a = st.slider('alpha', 0.0, 5.0, 2.5, 0.5)
+    s = st.slider('scale', 0.0, 5.0, 2.5, 0.5)
+
+    cdf = r'''
+                        cdf  = $\int_a^b f(x)dx$ 
+                        '''
+
+    st.latex(cdf)
+    q = tfd.Pareto(a, s)
+    z_values = tf.linspace(0, 5, 200)
+    z_values = tf.cast(z_values, tf.float32)
+    prob_values_p = p.prob(z_values)
+    prob_values_q = q.prob(z_values)
+
+    fig, ax = plt.subplots()
+    ax.plot(z_values, prob_values_p, label=r'p', linestyle='--', lw=2, alpha=0.5)
+    ax.plot(z_values, prob_values_q, label=r'q')
+
+    ax.set_xlabel("x")
+    ax.set_ylabel("PDF(x)")
+    ax.legend()
+
+    st.pyplot(fig)
+    kl = tfd.kl_divergence(q, p)
+    st.latex(f"D_{{KL}}(q||p) \\text{{  is : }}{kl:0.2f}")
+
+def Weibull():
+    st.header("Weibull distribution")
+    p = tfd.Weibull(1,2)
+    k = st.slider('k', 0.0, 5.0, 0.5, 0.5)
+    l = st.slider('lambda', 0.0, 5.0, 0.5, 0.5)
+
+    cdf = r'''
+                    cdf  = $\int_a^b f(x)dx$ 
+                    '''
+
+    st.latex(cdf)
+    q = tfd.Weibull(k, l)
+    z_values = tf.linspace(0, 10, 200)
+    z_values = tf.cast(z_values, tf.float32)
+    prob_values_p = p.prob(z_values)
+    prob_values_q = q.prob(z_values)
+
+    fig, ax = plt.subplots()
+    ax.plot(z_values, prob_values_p, label=r'p', linestyle='--', lw=5, alpha=0.5)
+    ax.plot(z_values, prob_values_q, label=r'q')
+
+    ax.set_xlabel("x")
+    ax.set_ylabel("PDF(x)")
+    ax.legend()
+    ax.set_ylim((0, 2))
+
+    st.pyplot(fig)
+    kl = tfd.kl_divergence(q, p)
+    st.latex(f"D_{{KL}}(q||p) \\text{{  is : }}{kl:0.2f}")
+
+    st.subheader("Relationship between Weibull and Exponential Distribution")
+    l1 = st.slider('l', 0.0, 5.0, 1.0, 0.5)
+    k1 = 1
+    st.write("k=1")
+    q2 = tfd.Weibull(k1, l1)
+    prob_values_q2 = list(q2.prob(z_values))
+    q3 = tfd.Exponential(1/l1)
+    prob_values_q3 = list(q3.prob(z_values))
+    fig3, ax3 = plt.subplots()
+    ax3.plot(z_values, prob_values_q2, linestyle='--', lw=3, alpha=0.5, label=r'Weibull(1,l)')
+    ax3.plot(z_values, prob_values_q3, label=r'Exponential(1/l)')
+
+    ax3.set_xlabel("x")
+    ax3.set_ylabel("PDF(x)")
+    ax3.legend()
+    ax3.set_ylim((0, 2))
+    st.pyplot(fig3)
+
+def StudentT():
+    st.subheader("StudentT Distribution")
+    p = tfd.StudentT(1,0,1)
+    df = st.slider('df',0,5,2,1)
+    loc = st.slider('loc', 0, 5, 2, 1)
+    scale = st.slider('scale', 0, 5, 2, 1)
+
+    cdf = r'''
+                        cdf  = $\int_a^b f(x)dx$ 
+                        '''
+
+    st.latex(cdf)
+    q = tfd.StudentT(df,loc,scale)
+    z_values = tf.linspace(-10, 10 , 200)
+    z_values = tf.cast(z_values, tf.float32)
+    prob_values_p = p.prob(z_values)
+    prob_values_q = q.prob(z_values)
+
+    fig, ax = plt.subplots()
+    ax.plot(z_values, prob_values_p, label=r'p', linestyle='--', lw=3, alpha=0.5)
+    ax.plot(z_values, prob_values_q, label=r'q')
+
+    ax.set_xlabel("x")
+    ax.set_ylabel("PDF(x)")
+    ax.legend()
+    ax.set_ylim((0, 1))
+
+    st.pyplot(fig)
+
+    st.subheader("Relationship between Cauchy and StudentT Distribution")
+    q2 = tfd.Cauchy(0, 1)
+    q3 = tfd.StudentT(1, 0, 1)
+    prob_values_q2 = list(q2.prob(z_values))
+    prob_values_q3 = list(q3.prob(z_values))
+
+    fig2, ax2 = plt.subplots()
+    ax2.plot(z_values, prob_values_q2, label=r'Cauchy(0,1)', linestyle='--', lw=3)
+    ax2.plot(z_values, prob_values_q3, label=r'StudentT(1,0,1)')
+
+    ax2.set_xlabel("x")
+    ax2.set_ylabel("PDF(x)")
+    ax2.legend()
+    ax2.set_ylim((0, 1))
+
+    st.pyplot(fig2)
+
+def Beta():
+    st.subheader("Beta distribution")
+    p = tfd.Beta(1.5,1.1)
+    alpha = st.slider('alpha', 0.0, 2.0, 1.2,0.1)
+    beta = st.slider('beta', 0.0, 2.0, 1.2,0.1)
+
+    cdf = r'''
+                    cdf  = $\int_a^b f(x)dx$ 
+                    '''
+
+    st.latex(cdf)
+    q = tfd.Beta(alpha, beta)
+    z_values = tf.linspace(0, 1, 200)
+    z_values = tf.cast(z_values, tf.float32)
+    prob_values_p = p.prob(z_values)
+    prob_values_q = q.prob(z_values)
+
+    fig, ax = plt.subplots()
+    ax.plot(z_values, prob_values_p, label=r'p', linestyle='--', lw=5, alpha=0.5)
+    ax.plot(z_values, prob_values_q, label=r'q')
+
+    ax.set_xlabel("x")
+    ax.set_ylabel("PDF(x)")
+    ax.legend()
+    #ax.set_ylim((0, 1))
+
+    st.pyplot(fig)
+    kl = tfd.kl_divergence(q, p)
+    st.latex(f"D_{{KL}}(q||p) \\text{{  is : }}{kl:0.2f}")
+
+    st.subheader("Relationship between Beta Distribution and Normal Distribution")
+    alpha1 = st.slider('beta=alpha', 50, 500,250, 50)
+    q2 = tfd.Beta(alpha1, alpha1)
+    q3 = tfd.Normal(0.5,sqrt(0.25/(2*alpha1+1)))
+    z_values1 = tf.linspace(0, 1, 200)
+    z_values1 = tf.cast(z_values1, tf.float32)
+    prob_values_q2 = q2.prob(z_values1)
+    prob_values_q3 = q3.prob(z_values1)
+
+    fig2, ax2 = plt.subplots()
+    ax2.plot(z_values1, prob_values_q2, label=r'Beta(alpha,Beta)', linestyle='--', lw=3)
+    ax2.plot(z_values1, prob_values_q3, label=r'Normal')
+
+    ax2.set_xlabel("x")
+    ax2.set_ylabel("PDF(x)")
+    ax2.legend()
+    #ax2.set_ylim((0, 1))
+
+    st.pyplot(fig2)
+
+    st.subheader("Relationship between Beta Distribution and Uniform Distribution")
+
+    st.write("beta = alpha= 1")
+    q4 = tfd.Beta(1,1)
+    q5 = tfd.Uniform(0,1)
+    z_values1 = tf.linspace(0, 1, 200)
+    z_values1 = tf.cast(z_values1, tf.float32)
+    prob_values_q4 = list(q4.prob(z_values1))
+    prob_values_q5 = list(q5.prob(z_values1))
+
+    fig3, ax3 = plt.subplots()
+    ax3.plot(z_values1, prob_values_q4, label=r'Beta(1,1)', linestyle='--', lw=3)
+    ax3.plot(z_values1, prob_values_q5, label=r'Uniform(0,1)')
+
+    ax3.set_xlabel("x")
+    ax3.set_ylabel("PDF(x)")
+    ax3.legend()
+    #ax3.set_ylim((0, 1))
+
+    st.pyplot(fig3)
+
+    st.subheader("Transformation of Beta Distribution")
+    alpha2 = st.slider('alpha2', 0.0, 2.0, 1.2,0.1)
+    beta2 = st.slider('beta2', 0.0, 2.0, 1.8,0.1)
+    p1 = tfd.Beta(alpha2, beta2)
+    prob_values_new = list(p1.prob(z_values))
+    z_values2 = np.zeros(len(z_values))
+    for i in range(len(z_values2)):
+        z_values2[i] = 1 - z_values[i]
+
+    q1 = tfd.Beta(beta2,alpha2)
+    new = list(q1.prob(z_values))
+    fig2, ax2 = plt.subplots()
+    ax2.plot(z_values, prob_values_new, label=r'X->Beta(alpha,beta)', lw=3)
+    ax2.plot(z_values2, prob_values_new, label=r'X transformed to 1-X', lw=2)
+    ax2.plot(z_values, new, label=r'X->Beta(beta,alpha)', linestyle='--', lw=2)
+
+    ax2.set_xlabel("x")
+    ax2.set_ylabel("PDF(x)")
+    ax2.legend()
+    st.pyplot(fig2)
+
+
+def Poisson():
+    st.subheader("Poisson distribution")
+    p = tfd.Poisson(5)
+    rate = st.slider('lambda', 0, 10, 1)
+
+    cdf = r'''
+                cdf  = $\int_a^b f(x)dx$ 
+                '''
+
+    st.latex(cdf)
+    q = tfd.Poisson(rate)
+    z_values = tf.linspace(-2, 10, 13)
+    z_values = tf.cast(z_values, tf.float32)
+    prob_values_p = p.prob(z_values)
+    prob_values_q = q.prob(z_values)
+
+    fig, ax = plt.subplots()
+    ax.stem(z_values, prob_values_p, label=r'Distribution with unknown parameters', linefmt='r', markerfmt='ro')
+    ax.stem(z_values, prob_values_q, label=r'Distribution with given parameters', linefmt='--', markerfmt='bo')
+
+    ax.set_xlabel("x")
+    ax.set_ylabel("PDF(x)")
+    ax.legend()
+    ax.set_ylim((0, 1))
+    st.pyplot(fig)
+    st.subheader("Addition of two Poisson distributions")
+    rate1 = st.slider('lambda1', 0, 10, 1)
+    rate2 = st.slider('lambda2', 0, 10, 1)
+    q1 = tfd.Poisson(rate1)
+    q2 = tfd.Poisson(rate2)
+    prob_values_q1 = list(q1.prob(z_values))
+    prob_values_q2 = list(q2.prob(z_values))
+    fig2, (ax2,ax3) = plt.subplots(1,2)
+    ax2.stem(z_values, prob_values_q1, linefmt='r', markerfmt='ro')
+    ax3.stem(z_values, prob_values_q2, linefmt='--', markerfmt='bo')
+
+    ax2.set_xlabel("x")
+    ax2.set_title("Poisson(lambda1)")
+    ax2.set_ylim((0, 1))
+
+    ax3.set_xlabel("x")
+    ax3.set_title("Poisson(lambda2)")
+    ax3.set_ylim((0, 1))
+
+    st.pyplot(fig2)
+
+    prob_values_sum =Sum(prob_values_q1, prob_values_q2, z_values)
+    q3 = tfd.Poisson(rate1+rate2)
+    prob_values_q3 = list(q3.prob(range(len(prob_values_sum))))
+    fig3, ax4 = plt.subplots()
+    ax4.stem(range(len(prob_values_sum)), prob_values_sum, label=r'Poisson(lambda1)+Poisson(lambda2)', linefmt='r', markerfmt='ro')
+    ax4.stem(range(len(prob_values_sum)), prob_values_q3, linefmt='--', label=r'Poisson(lambda1+lambda2)', markerfmt='bo')
+    ax4.set_xlabel("x")
+    ax4.set_ylabel("PDF(x)")
+    ax4.legend()
+    ax4.set_ylim((0, 1))
+    st.pyplot(fig3)
+    st.markdown("Summation of two poisson distribution with parameter lamba1 and lambda2 yields Poisson distribution with parameter (lambda1+lambda2)")
+
+    st.subheader("Relationship between Poisson and Normal Distribution")
+    rate3 = st.slider('lambda1', 100, 500,250,50)
+    q4 = tfd.Poisson(rate=rate3)
+    z_values1 = tf.linspace(0, 600, 601)
+    z_values1 = tf.cast(z_values1, tf.float32)
+    prob_values_q4 = list(q4.prob(z_values1))
+    q5 = tfd.Normal(rate3, sqrt(rate3))
+    prob_values_q5 = list(q5.prob(z_values1))
+
+    fig4, ax5 = plt.subplots()
+    ax5.stem(z_values1, prob_values_q4, label=r'Poisson(lambda1)]', linefmt='r', markerfmt='ro')
+    ax5.plot(z_values1, prob_values_q5, label=r'Normal(lamda1,sqrt(lambda1)', lw=3)
+    ax5.set_xlabel("x")
+    ax5.set_ylabel("PDF(x)")
+    ax5.legend()
+    ax5.set_ylim((0, 0.1))
+    st.pyplot(fig4)
+    st.markdown("for large values of lambda, Poisson(lambda) becomes approximately a normal distribution having mean= lambda and variance= lambda")
+
+
+def Binomial():
+    st.subheader("Binomial distribution")
+    p = tfd.Binomial(total_count=5, probs=.5)
+    count = st.slider('n', 1, 10, 4, 1)
+    prob = st.slider('prob', 0.0, 1.0, 0.5,0.1)
+    st1 = r'''
+                 Mean = \[n p \]
+                Variance = \[n p q\]
+                 Entropy = \[\frac{1}{2} \log_2 (2 \pi n e p q)\ + O ( \frac {1}{n}) \]
+                '''
+
+    st.latex(st1)
+    q = tfd.Binomial(total_count=count, probs=prob )
+    z_values = tf.linspace(0, 10, 11)
+    z_values = tf.cast(z_values, tf.float32)
+    prob_values_p = list(p.prob(z_values))
+    prob_values_q = list(q.prob(z_values))
+
+    fig, ax = plt.subplots()
+    ax.stem(z_values, prob_values_p, label=r'Distribution with unknown parameters', linefmt = 'r', markerfmt = 'ro')
+    ax.stem(z_values, prob_values_q, label=r'Distribution with given parameters', linefmt ='--', markerfmt = 'bo')
+
+    ax.set_xlabel("x")
+    ax.set_ylabel("PDF(x)")
+    ax.legend()
+    ax.set_ylim((0, 1))
+
+    st.pyplot(fig)
+
+    st.markdown("Relationship between Poisson and Binomial Distribution")
+    count1 = st.slider('n', 750, 1000, 800, 25)
+    prob1 = st.slider('prob', 0.0, 0.010, 0.001, 0.001)
+
+    t = tfd.Binomial(total_count=count1, probs=prob1)
+    r = tfd.Poisson(count1*prob1)
+
+    z_values1 = tf.linspace(0, 50, 51)
+    z_values1 = tf.cast(z_values1, tf.float32)
+    prob_values_t = t.prob(z_values1)
+    prob_values_r = r.prob(z_values1)
+
+    fig1, ax = plt.subplots()
+    ax.stem(z_values1, prob_values_t, label=r'binomial', linefmt='r', markerfmt='ro')
+    ax.stem(z_values1, prob_values_r, label=r'poisson', linefmt='--', markerfmt='bo')
+
+    ax.set_xlabel("x")
+    ax.set_ylabel("PDF(x)")
+    ax.legend()
+    ax.set_ylim((0, 0.5))
+
+    st.pyplot(fig1)
+    st.markdown("")
+    st.markdown("It is clear from the graph, that for large values of n and small values of p, the binomial distribution approximates to poisson distribution.")
+    st.markdown("")
+
+    st.markdown("Transformation of Binomial Distribution")
+    count2 = st.slider('n1', 1, 10, 4, 1)
+    prob2 = st.slider('p', 0.0, 1.0, 0.5, 0.1)
+    p1 = tfd.Binomial(total_count=count2, probs=prob2)
+    prob_values_n_p = list(p1.prob(z_values))
+    z_values2=np.zeros(len(z_values))
+    for i in range(len(z_values2)):
+        z_values2[i]=count2-z_values[i]
+
+    q1 = tfd.Binomial(total_count=count2, probs=1-prob2)
+    new = list(q1.prob(z_values))
+    fig2, ax = plt.subplots()
+    ax.stem(z_values, prob_values_n_p, label=r'X->Binomial(n,p)', linefmt='r', markerfmt='ro')
+    ax.stem(z_values2, prob_values_n_p, label=r'X transformed to N-X', linefmt='g', markerfmt='go')
+    ax.stem(z_values, new, label=r'X->Binomial(n,1-p)', linefmt='--', markerfmt='bo')
+
+    ax.set_xlabel("x")
+    ax.set_ylabel("PDF(x)")
+    ax.legend()
+    ax.set_ylim((0, 1))
+    ax.set_xlim((-0.5,count2+0.5))
+    st.pyplot(fig2)
+    st.markdown("When a r.v. X with the binomial(n,p) distribution is transformed to n-X, we get a binomial(n,1-p) distribution.")
+
+    st.markdown("Relationship between Normal and Binomial Distribution")
+    count3 = st.slider('n3', 750, 1000, 800, 25)
+    prob3 = st.slider('prob3', 0.0, 1.0, 0.5, 0.1)
+
+    t = tfd.Binomial(total_count=count3, probs=prob3)
+    r = tfd.Normal(count3 * prob3, sqrt((count3*prob3*(1-prob3))))
+
+    z_values1 = tf.linspace(0, 1000, 1001)
+    z_values1 = tf.cast(z_values1, tf.float32)
+    prob_values_t = t.prob(z_values1)
+    prob_values_r = r.prob(z_values1)
+
+    fig2, ax2 = plt.subplots()
+    ax2.stem(z_values1, prob_values_t, label=r'Binomial', linefmt='r', markerfmt='ro')
+    ax2.plot(z_values1, prob_values_r, label=r'Normal', lw=3)
+
+    ax2.set_xlabel("x")
+    ax2.set_ylabel("PDF(x)")
+    ax2.legend()
+    ax2.set_ylim((0, 0.1))
+
+    st.pyplot(fig2)
+    st.markdown("")
+    st.markdown("For large values of n, the binomial distribution approximates to the normal distribution with mean = n*p and variance = n*p*(1-p)")
+    st.markdown("")
+
+
+def Bernoulli_dist():
+    st.header("Bernoulli distribution")
+    p = tfd.Bernoulli(probs=0.5)
+    suc = st.slider('p', 0.0, 1.0, 0.8, 0.1)
+
+    pmf = r'''
+                
+                pmf  = f(x) = p  \quad  \qquad if x=1
+                           \\ \qquad \qquad \quad \,\,\,= 1-p \quad   \,\, if x=0
+                           \\ \,\,\,\quad\qquad= 0     \quad\qquad  else
+                '''
+    mean = suc
+    variance = suc*(1-suc)
+
+    st1 = f'''
+            mean = p = {mean}\\\\
+            variance = p*(1-p) = {variance:0.3f}\\\\
+            Entropy = -q lnq - p ln p
+            '''
+    st.latex(pmf)
+    st.latex(st1)
+    q = tfd.Bernoulli(probs = suc)
+    z_values = tf.linspace(0, 1, 2)
+    z_values = tf.cast(z_values, tf.float32)
+    prob_values_p = p.prob(z_values)
+    prob_values_q = q.prob(z_values)
+    fig, ax = plt.subplots()
+    ax.stem(z_values, prob_values_p, label=r'Distribution with unknown parameters', linefmt='b', markerfmt='bo')
+    ax.stem(z_values, prob_values_q, label=r'Distribution with given parameters', linefmt='g', markerfmt='go')
+
+    ax.set_xlabel("x")
+    ax.set_ylabel("PDF(x)")
+    ax.legend()
+    ax.set_ylim((0, 1))
+
+    st.pyplot(fig)
+    st.subheader("Multiplication of two Bernoulli distributions")
+    p1 = st.slider('p1', 0.0, 1.0, 0.2, 0.1)
+    p2 = st.slider('p2', 0.0, 1.0, 0.7, 0.1)
+
+    rv_p1 = tfd.Bernoulli(probs=p1)
+    rv_p2 = tfd.Bernoulli(probs=p2)
+    prob_values_p1 = rv_p1.prob(z_values)
+    prob_values_p2 = rv_p2.prob(z_values)
+    prob_values_pone = list(prob_values_p1)
+    prob_values_ptwo = list(prob_values_p2)
+    prob_values_pdt = Product(prob_values_pone, prob_values_ptwo, z_values)
+
+    fig1,(ax1,ax2)=plt.subplots(1,2)
+    ax1.stem(z_values, prob_values_p1, label=r'p1', linefmt='b', markerfmt='bo')
+    ax2.stem(z_values, prob_values_p2, label=r'p2', linefmt='g', markerfmt='go')
+
+    ax1.set_title("Bernoulli(p1)")
+    ax1.set_xlabel("x")
+    ax1.set_ylabel("PDF(x)")
+    ax1.set_ylim((0, 1))
+
+    ax2.set_title("Bernoulli(p2)")
+    ax2.set_xlabel("x")
+    ax2.set_ylim((0, 1))
+
+    st.pyplot(fig1)
+
+    rv_p1p2 = tfd.Bernoulli(probs=p1*p2)
+    prob_values_p1p2 = rv_p1p2.prob(range(len(prob_values_pdt)))
+
+    fig2, ax3 = plt.subplots()
+    ax3.stem(range(len(prob_values_pdt)), prob_values_pdt, label=r'Product of Bernoulli(p1) and Bernoulli(p2)', linefmt='r', markerfmt='ro')
+    ax3.stem(range(len(prob_values_pdt)), prob_values_p1p2, label=r'Bernoulli(p1*p2)', linefmt='--', markerfmt='bo')
+    ax3.set_xlabel("x")
+    ax3.set_ylabel("PDF(x)")
+    ax3.legend()
+    ax3.set_ylim((0, 1))
+    ax3.set_xlim((-0.5, 1.5))
+
+    st.pyplot(fig2)
+    st.markdown("Multiplication of a Bernoulli(p1) distribution with Bernouli(p2) gives a Bernoulli distribution with parameter=p1*p2")
+    st.subheader("Addition of two Bernoulli distributions")
+    p3 = st.slider('p3', 0.0, 1.0, 0.6, 0.1)
+    rv_p3 = tfd.Bernoulli(probs=p3)
+    prob_values_p3 = list(rv_p3.prob(z_values))
+    prob_values_sum=Sum(prob_values_p3, prob_values_p3, z_values)
+    fig3, ax4 = plt.subplots()
+    ax4.stem(range(len(prob_values_sum)), prob_values_sum, label=r'Sum of 2 Bernoulli(p3) distributions', linefmt='r', markerfmt='ro')
+    b = tfd.Binomial(total_count=2, probs=p3)
+    prob_values_bin = list(b.prob(range(len(prob_values_sum))))
+    ax4.stem(range(len(prob_values_sum)), prob_values_bin, label=r'Binomial(2,p3)', linefmt='--', markerfmt='bo')
+    ax4.set_xlabel("x")
+    ax4.set_ylabel("PDF(x)")
+    ax4.legend()
+    ax4.set_ylim((0, 1))
+
+    st.pyplot(fig3)
+    st.markdown("The sum of n Bernoulli(p) distributions is a binomial(n,p) distribution")
+
+def BetaBinomial():
+    st.markdown("Beta Binomial distribution")
+    p = tfd.BetaBinomial(8,0.5,1.2)
+    num = st.slider('n', 0, 10, 5, 1)
+    al = st.slider('alpha', 0.0, 5.0, 0.2, 0.1)
+    be = st.slider('beta', 0.0, 5.0, 0.2, 0.1)
+
+
+    st1 = r'''
+                Mean = \[ \frac {n \alpha}{\alpha + \beta} \]
+                Variance =  \[ \frac {(n \alpha \beta)(\alpha + \beta + n)}{(\alpha + \beta + 1) (\alpha + \beta)^2} \]
+                '''
+
+    st.latex(st1)
+    q = tfd.BetaBinomial(num, al, be)
+    z_values = tf.linspace(0, 10, 11)
+    z_values = tf.cast(z_values, tf.float32)
+    prob_values_p = p.prob(z_values)
+    prob_values_q = q.prob(z_values)
+
+    fig, ax = plt.subplots()
+    ax.stem(z_values, prob_values_p, label=r'Distribution with unknown parameters', linefmt='r', markerfmt='ro')
+    ax.stem(z_values, prob_values_q, label=r'Distribution with given parameters', linefmt='--', markerfmt='bo')
+
+    ax.set_xlabel("x")
+    ax.set_ylabel("PDF(x)")
+    ax.legend()
+    ax.set_ylim((0, 1))
+
+    st.pyplot(fig)
+
+    st.markdown("")
+    st.markdown("Relarionship between BetaBinomial and Uniform-Discrete Distribution")
+    st.markdown("")
+    num2 = st.slider('n2', 0, 10, 5, 1)
+    q2 = tfd.BetaBinomial(num2, 1, 1)
+    prob_values_q2 = q2.prob(z_values)
+
+
+    fig2, ax2 = plt.subplots()
+    ax2.stem(z_values, prob_values_q2, label=r'BetaBinomial(n,1,1)', linefmt='r', markerfmt='ro')
+
+    ax2.set_xlabel("x")
+    ax2.set_ylabel("PDF(x)")
+    ax2.legend()
+    ax2.set_ylim((0, 1))
+
+    st.pyplot(fig2)
+    st.markdown("A BetaBinomial(n,alpha,beta) with alpha=beta=1 becomes a uniform-discrete distribution from 0 to n(n2 here)")
+    st.markdown("")
+
+    st.markdown("Relationship between Binomial and BetaBinomial Distribution")
+    num3 = st.slider('_n', 0, 10, 5, 1)
+    al2 = st.slider('_alpha', 100, 500, 200, 50)
+    be2 = st.slider('_beta', 100, 500, 200, 50)
+
+    q3 = tfd.BetaBinomial(num3, al2, be2)
+    prob_values_q3 = q3.prob(z_values)
+
+    success = al2/(al2+be2)
+    q4 = tfd.Binomial(total_count=num3, probs=success)
+    prob_values_q4 = q4.prob(z_values)
+
+    fig3, ax3 = plt.subplots()
+    ax3.stem(z_values, prob_values_q3, label=r'BetaBinomial', linefmt='r', markerfmt='ro')
+    ax3.stem(z_values, prob_values_q4, label=r'Binomial', linefmt='--', markerfmt='bo')
+    ax3.set_xlabel("x")
+    ax3.set_ylabel("PDF(x)")
+    ax3.legend()
+    ax3.set_ylim((0, 1))
+
+    st.pyplot(fig3)
+    st.markdown("For large values of (alpha+beta), the BetaBinomial approximates to the Binomial Distribution with the probanility of success = alpha/(alpha+beta)")
+    st.markdown("")
+
+def Geometric():
+    st.subheader("Geometric distribution")
+    p = tfd.Geometric(probs=0.2)
+    prob = st.slider('prob', 0.0, 1.0, 0.4, 0.1)
+
+    st1 = r'''
+                Mean=\[ \frac {1}{p} \]
+                Variance= \[ \frac {1-p}{p^2} \]
+                Entropy = \[ \frac { -(1-p) \log_2 {(1-p)} - p \log_2 p} {p}\]
+                '''
+
+    st.latex(st1)
+    q = tfd.Geometric(probs=prob)
+    z_values = tf.linspace(0, 10, 11)
+    z_values = tf.cast(z_values, tf.float32)
+    prob_values_p = p.prob(z_values)
+    prob_values_q = q.prob(z_values)
+
+    fig, ax = plt.subplots()
+    ax.stem(z_values, prob_values_p, label=r'Distribution with unknown parameters', linefmt='b', markerfmt='bo')
+    ax.stem(z_values, prob_values_q, label=r'Distribution with given parameters', linefmt='g', markerfmt='go')
+    ax.set_xlabel("x")
+    ax.set_ylabel("PMF(x)")
+    ax.legend()
+    ax.set_ylim((0, 1))
+
+    st.pyplot(fig)
+
+def Gamma():
+    st.subheader("Gamma distribution")
+    p = tfd.Gamma(concentration = 3.5, rate = 3 )
+    concn = st.slider('concentration', 0.0, 10.0, 5.0, 0.5)
+    rat = st.slider('rate', 0.0, 10.0, 2.0, 0.5)
+
+    cdf = r'''
+                cdf  = $\int_a^b f(x)dx$ 
+                '''
+
+    st.latex(cdf)
+    q = tfd.Gamma(concentration = concn, rate = rat )
+    z_values = tf.linspace(0, 20, 200)
+    z_values = tf.cast(z_values, tf.float32)
+    prob_values_p = p.prob(z_values)
+    prob_values_q = q.prob(z_values)
+
+    fig, ax = plt.subplots()
+    ax.plot(z_values, prob_values_p, label=r'p', linestyle='--', lw=5, alpha=0.5)
+    ax.plot(z_values, prob_values_q, label=r'q')
+
+    ax.set_xlabel("x")
+    ax.set_ylabel("PDF(x)")
+    ax.legend()
+    #ax.set_ylim((0, 1))
+
+    st.pyplot(fig)
+    kl = tfd.kl_divergence(q, p)
+    st.latex(f"D_{{KL}}(q||p) \\text{{  is : }}{kl:0.2f}")
+
+    st.subheader("Relationship between Gamma and Exponential Distribution")
+    alpha1=1
+    st.write("alpha1=1")
+    beta1 = st.slider('beta1', 0.0, 10.0, 5.0, 0.5)
+    q2 = tfd.Gamma(concentration=alpha1, rate=beta1)
+    prob_values_q2 = list(q2.prob(z_values))
+    q3 = tfd.Exponential(beta1)
+    prob_values_q3 = list(q3.prob(z_values))
+    fig3, ax3 = plt.subplots()
+    ax3.plot(z_values, prob_values_q2, linestyle='--', lw=3, alpha=0.5, label=r'Gamma(1,beta)')
+    ax3.plot(z_values, prob_values_q3, label=r'Exponential(beta)')
+
+    ax3.set_xlabel("x")
+    ax3.set_ylabel("PDF(x)")
+    ax3.legend()
+    #ax3.set_ylim((0, 1))
+    st.pyplot(fig3)
+
+def NegBin():
+    st.markdown("Negative Binomial distribution")
+    p = tfd.NegativeBinomial(total_count=5, probs=.5)
+    count = st.slider('n', 1, 10, 1)
+    prob = st.slider('prob', 0.0, 1.0, 0.1)
+    cdf = r'''
+                cdf  = $\int_a^b f(x)dx$ 
+                '''
+
+    st.latex(cdf)
+    q = tfd.NegativeBinomial(total_count=count, probs=prob )
+    z_values = tf.linspace(0, 100, 100)
+    z_values = tf.cast(z_values, tf.float32)
+    prob_values_p = p.prob(z_values)
+    prob_values_q = q.prob(z_values)
+
+    fig, ax = plt.subplots()
+    ax.stem(z_values, prob_values_p, label=r'Distribution with unknown parameters', linefmt='b', markerfmt='bo')
+    ax.stem(z_values, prob_values_q, label=r'Distribution with given parameters', linefmt='g', markerfmt='go')
+
+    ax.set_xlabel("x")
+    ax.set_ylabel("PMF(x)")
+    ax.legend()
+    ax.set_ylim((0, 1))
+
+    st.pyplot(fig)
+
+
+
+if (add_selectbox == "Continuous Univariate"):
+    selection1 = st.sidebar.selectbox(
+        'Choose an Option',
+        ('Beta','Cauchy', 'Chi', 'Chi-Squared', 'Exponential', 'Gamma', 'Laplace', 'Normal',
+         'Pareto', 'StudentT', 'Uniform', 'Weibull')
+    )
+    if (selection1 == "Normal"):
+        Normal()
+    elif (selection1 == "Exponential"):
+        Exponential()
+    elif(selection1 == "Uniform"):
+        Uniform()
+    elif(selection1 == "Error"):
+        Error()
+    elif (selection1 == "Cauchy"):
+        Cauchy()
+    elif (selection1 == "Chi"):
+        Chi()
+    elif (selection1 == "Chi-Squared"):
+        Chi_squared()
+    elif (selection1 == "Laplace"):
+        Laplace()
+    elif(selection1=="Pareto"):
+        Pareto()
+    elif(selection1 == "Weibull"):
+        Weibull()
+    elif (selection1 == "Gamma"):
+        Gamma()
+    elif (selection1 == "StudentT"):
+        StudentT()
+    else:
+        Beta()
+elif (add_selectbox == "Discrete Univariate"):
+    selection1 = st.sidebar.selectbox(
+        'Choose an Option',
+        ('Bernoulli', 'Beta-Binomial','Binomial', 'Geometric', 'Negative-Binomial', 'Poisson')
+    )
+    if (selection1 == "Poisson"):
+        Poisson()
+    elif (selection1 == "Bernoulli"):
+        Bernoulli_dist()
+    elif (selection1 == "Binomial"):
+        Binomial()
+    elif (selection1 == "Beta-Binomial"):
+        BetaBinomial()
+    elif (selection1 == "Geometric"):
+        Geometric()
+    elif (selection1 == "Negative-Binomial"):
+        NegBin()